Learning Data Manifolds with a Cutting Plane Method
This work addresses the challenge of learning from data manifolds for machine learning practitioners, offering a more efficient alternative to data augmentation.
The paper tackles the problem of classifying data manifolds with continuous invariances by proposing an iterative cutting-plane algorithm, M_{CP}, which efficiently finds maximum margin solutions and demonstrates superior generalization performance compared to conventional data augmentation methods.
We consider the problem of classifying data manifolds where each manifold represents invariances that are parameterized by continuous degrees of freedom. Conventional data augmentation methods rely upon sampling large numbers of training examples from these manifolds; instead, we propose an iterative algorithm called M_{CP} based upon a cutting-plane approach that efficiently solves a quadratic semi-infinite programming problem to find the maximum margin solution. We provide a proof of convergence as well as a polynomial bound on the number of iterations required for a desired tolerance in the objective function. The efficiency and performance of M_{CP} are demonstrated in high-dimensional simulations and on image manifolds generated from the ImageNet dataset. Our results indicate that M_{CP} is able to rapidly learn good classifiers and shows superior generalization performance compared with conventional maximum margin methods using data augmentation methods.